Properties

Label 2-177-59.58-c2-0-19
Degree $2$
Conductor $177$
Sign $-0.486 - 0.873i$
Analytic cond. $4.82290$
Root an. cond. $2.19611$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.07i·2-s + 1.73·3-s − 5.44·4-s − 6.03·5-s − 5.32i·6-s − 9.30·7-s + 4.42i·8-s + 2.99·9-s + 18.5i·10-s + 12.6i·11-s − 9.42·12-s − 16.7i·13-s + 28.5i·14-s − 10.4·15-s − 8.16·16-s + 14.0·17-s + ⋯
L(s)  = 1  − 1.53i·2-s + 0.577·3-s − 1.36·4-s − 1.20·5-s − 0.886i·6-s − 1.32·7-s + 0.553i·8-s + 0.333·9-s + 1.85i·10-s + 1.15i·11-s − 0.785·12-s − 1.29i·13-s + 2.04i·14-s − 0.696·15-s − 0.510·16-s + 0.828·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.486 - 0.873i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.486 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.486 - 0.873i$
Analytic conductor: \(4.82290\)
Root analytic conductor: \(2.19611\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :1),\ -0.486 - 0.873i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.264780 + 0.450313i\)
\(L(\frac12)\) \(\approx\) \(0.264780 + 0.450313i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 1.73T \)
59 \( 1 + (28.6 + 51.5i)T \)
good2 \( 1 + 3.07iT - 4T^{2} \)
5 \( 1 + 6.03T + 25T^{2} \)
7 \( 1 + 9.30T + 49T^{2} \)
11 \( 1 - 12.6iT - 121T^{2} \)
13 \( 1 + 16.7iT - 169T^{2} \)
17 \( 1 - 14.0T + 289T^{2} \)
19 \( 1 + 37.4T + 361T^{2} \)
23 \( 1 + 27.8iT - 529T^{2} \)
29 \( 1 - 39.1T + 841T^{2} \)
31 \( 1 + 0.257iT - 961T^{2} \)
37 \( 1 + 54.7iT - 1.36e3T^{2} \)
41 \( 1 + 11.3T + 1.68e3T^{2} \)
43 \( 1 + 7.36iT - 1.84e3T^{2} \)
47 \( 1 + 28.7iT - 2.20e3T^{2} \)
53 \( 1 + 47.3T + 2.80e3T^{2} \)
61 \( 1 - 84.0iT - 3.72e3T^{2} \)
67 \( 1 - 81.5iT - 4.48e3T^{2} \)
71 \( 1 + 6.96T + 5.04e3T^{2} \)
73 \( 1 + 131. iT - 5.32e3T^{2} \)
79 \( 1 - 51.9T + 6.24e3T^{2} \)
83 \( 1 + 16.5iT - 6.88e3T^{2} \)
89 \( 1 + 69.2iT - 7.92e3T^{2} \)
97 \( 1 - 110. iT - 9.40e3T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.12226426392851292917652585169, −10.54003366544853828924969998863, −10.21530294266856997900885926019, −9.011857893904726975012487072883, −7.938413753473083368976667468298, −6.65818802589527440009490459041, −4.43483860147305859758152854994, −3.53159282966139616836478645978, −2.51659899709513722158726777305, −0.28283939527970848815174374317, 3.31559950659632487811801839556, 4.42594851086345054511893018671, 6.18137491448774075334037959856, 6.82886035389415948474612681835, 8.005593486917268674650619926320, 8.626874352763587641192739665585, 9.661935244253918114463096773661, 11.20743383389067886920293642167, 12.36272842444627324945798465994, 13.51334513036215786429844999100

Graph of the $Z$-function along the critical line